Components of VectorsSection 4.2

Physics

Components of Vectors

• Breaking vectors into their component parts: Vector Resolution

• The magnitude and sign of component vectors: Components

• Components are found using trigonometry.

Component Vectors

Θ

x

y

A

Ax

Ay

Component Vectors

• Ay = A sin Θ

– Sin = Opposite / Hypotenuse (SOH)

• Ax = A cos Θ

– Cos = Adjacent / Hypotenuse (CAH)

Vector Quadrants

• The sign of a component depends upon which of the four quadrants the component is in.

• When the angle that a vector makes with the x-axis is larger than 90° -- the sign of one or more components is negative.

Ax > 0

Ay > 0

Both Components are positive

First QuadrantSecond Quadrant

Third Quadrant Fourth Quadrant

Ax < 0

Ay > 0

Ax < 0

Ay < 0

Ax > 0

Ay < 0

x

y

Example Problem

• A bus travels 23.0 km on a straight road that is 30° north of east. What are the east and north components of its displacement? N

E

S

W

A

Ax

Ay

Θ

Problems

• What are the components of a vector of magnitude 1.5 m at an angle of 35° from the positive x-axis?

• A hiker walks 14.7 km at an angle 35° south of east. Find the east and north components of this walk.

More Problems

• An airplane flies at 65 m/s in the direction 149° counterclockwise from east. What are the east and north components of the plane’s velocity?

• A golf ball, hit from the tee, travels 325 m in a direction 25° south of the east axis. What are the east and north components of its displacement?

Not All Things Travel in a Straight Line!

• Algebraic Addition of Vectors:– Two or more vectors can be divided into their

x and y components and the resultant vector can be found.

Resultant Vector

A

B

C

Resultant VectorR² = Rx² + Ry²

x

y

Ax Bx Cx

Ay

By

Cy

Ry = Ay + By + Cy

Rx = Ax + Bx + Cx

Resultant Angle

Θ = ?

x

y

R

Rx

Ry

tan Θ = Rx² ÷ Ry²tan = TOA

Problems

• A powerboat heads due northwest at 13 m/s with respect to the water across a river that flows due north at 5 m/s. What is the velocity (both magnitude and direction) of the motorboat with respect to the shore?

Problems

• An airplane flies due south at 175 km/h with respect to the air. There is a wind blowing at 85 km/h to the east relative to the ground. What are the plane’s speed and direction with respect to the ground?

Problems

• An airplane flies due north at 235 km/h with respect to the air. There is a wind blowing at 65 km/h to the northeast with respect to the ground. What are the plane’s speed and direction with respect to the ground?

The End